Signals as Vectors

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Signals as Vectors

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.
​

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page  TOC  INDEX  Next page >>

Signals as Vectors

For the DFT, all signals and spectra are length $N$. A length $N$ sequence$x$ can be denoted by $x(n)$, $n=0,1,2,\, where $x(n)$ may be real ($x\) or complex ($x\). We now wish to regard $x$ as avector $\6.1 in an $N$ dimensional vector space. That is, each sample $x(n)$ is regarded as a coordinate in that space. A vector $\ is mathematically a single point in $N$-space represented by a list of coordinates $(x_0,x_1,x_2,\ called an $N$-tuple. (The notation $x_n$ means the same thing as $x(n)$.) It can be interpreted geometrically as an arrow in $N$-space from the origin $\ to the point $\.

We define the following as equivalent:

\

where $x_n \ is the $n$th sample of the signal (vector) $x$. From now on, unless specifically mentioned otherwise, all signals are length $N$.


Subsections

<< Previous page  TOC  INDEX  Next page >>

​
 
​
© 1998-2023 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy